A rectangular sheet of paper 30 into 18 cm square can be transformed into the curved surface of a right circular cylinder in two ways that is either by Rolling the paper along its length of a rolling it along its breadth
find the ratio of the volume of the two cylinders does formed
case 1: the circumference of the cylinder is 30, and the height is 18
2?r = 30
r = 15/?
V = ?(15/?)^2 (18) = 4050/?
case 2: the circumference is 18 and the height is 30
2?r = 18
r = 9/?
V = ?(9/?)^2 (30) = 2430/?
volume of case1 : volume of case2
= 4050/? : 2430/?
= 5 : 3
Just realized that I answered the same question back in 2013 and got the same answer,
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To find the ratio of the volumes of the two cylinders formed, we need to determine the dimensions of the cylinders.
1. Rolling the paper along its length:
When the paper is rolled along its length, the circumference of the base of the cylinder will be equal to the length of the paper, which is 30 cm. The height of the cylinder will be the breadth of the paper, which is 18 cm. Let's denote this cylinder as Cylinder 1.
2. Rolling the paper along its breadth:
When the paper is rolled along its breadth, the circumference of the base of the cylinder will be equal to the breadth of the paper, which is 18 cm. The height of the cylinder will be the length of the paper, which is 30 cm. Let's denote this cylinder as Cylinder 2.
Now we can calculate the volumes of the two cylinders using the formula for the volume of a cylinder.
Volume of Cylinder = π * (radius)^2 * height
For Cylinder 1:
- Radius: The radius of the base of Cylinder 1 is half the circumference, which is 30 cm / (2π) = 15 cm/(2 * π).
- Height: The height of Cylinder 1 is 18 cm.
So, the volume of Cylinder 1 is:
Volume 1 = π * (15 cm/(2*π))^2 * 18 cm
For Cylinder 2:
- Radius: The radius of the base of Cylinder 2 is half the circumference, which is 18 cm / (2π) = 9 cm/(2 * π).
- Height: The height of Cylinder 2 is 30 cm.
So, the volume of Cylinder 2 is:
Volume 2 = π * (9 cm/(2*π))^2 * 30 cm
Now, we can calculate the ratio of the volumes:
Ratio = Volume 1 / Volume 2
= (π * (15 cm/(2*π))^2 * 18 cm) / (π * (9 cm/(2*π))^2 * 30 cm)
= (15 cm/(2*π))^2 * 18 cm / (9 cm/(2*π))^2 * 30 cm
Simplifying further, we get:
Ratio = (15/9)^2 * (18/30)
= (5/3)^2 * (3/5)
= 25/9
Therefore, the ratio of the volumes of the two cylinders is 25:9.